A differential model of controlled cardiac pacemaker cell
Karima Djabella, Michel Sorine
To cite this version:
Karima Djabella, Michel Sorine. A differential model of controlled cardiac pacemaker cell. 6th
IFAC Symposium on Modelling and Control in Biomedical Systems, Sep 2006, Reims, 2006.
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A DIFFERENTIAL MODEL OF CONTROLLED
CARDIAC PACEMAKER CELL
Karima Djabella ∗ Michel Sorine ∗
∗
INRIA Rocquencourt / B.P.105.78153 Le Chesnay Cedex,
France
Abstract: A differential model of a cardiac pacemaker cell with only ten state
variables is proposed. It is intended for 0D or 3D simulation of the heart under
the vagal control of the autonomous nervous system. Three variables are used
to describe the membrane (membrane potential and two gate variables of ionic
channels), taking into account the dynamics of the main ionic currents (inward
sodium, L-type calcium and outward potassium), N a+ /Ca2+ exchangers and
N a+ /K + pumps. The remaining seven variables are associated with the fluid
compartment model that includes Ca2+ binding by myoplasmic proteins, and
the intracellular concentrations of free Calcium, Sodium and Potassium. Despite
its moderate number of state variables, this model includes the main processes
thought to be important in pacemaking on the cell scale and predicts the
experimentally observed ionic concentration of calcium, sodium and potassium,
action potential and membrane currents. The control by the calcium of the
c
pacemaking activity is also considered. Copyright 2006
IFAC
Keywords: Electrical activity, Nonlinear systems, Dynamic modelling, Frequency
control.
1. INTRODUCTION
There are many mathematical models of the electrophysiology of the different types of cardiac
cells. For models of human ventricular or atrial
cells, see e.g. (Tusscher et al., 2004; Nygren et
al., 1998) and the references herein. The sinoatrial
node cells and their pacemaker activity, considered here, have been studied in (Demir et al.,
1994; Dokos et al., 1996; Dokos et al., 1998; Zhang
et al., 2000; Kurata et al., 2002). These models
result from iterations between mathematics and
experimentation on the cell scale where detailed
characteristics of isolated ion channels can now
be measured. They are based on the ionic current
model of (Hodgkin and Huxley, 1952), but the
cardiac myocytes being far more complex than the
squid giant axon considered by this first model,
their complexity is high with e.g. 28 state variables
for the model in (Kurata et al., 2002).
Recently, model-based image and signal processing on the heart scale has become an important
goal as in (CardioSense3D, 2006). In such project,
models are not only needed in direct computations
to gain insights and for their predictive capabilities, but also in inverse problems to estimate state
and parameters from measurements. In that case
it is necessary to predict the shape of action potential (AP) for electrocardiogram interpretation,
as well as the concentration of calcium bound
on Troponin C, responsible for electromechanical
coupling at the origin of heart deformations seen
in the images. It is then necessary to represent
also some intracellular calcium buffering, as will
be done here, a special attention being paid to
model complexity in order to have a good tradeoff
between the descriptive power of the model and
the well-posedness of associated inverse problems.
The ionic currents that control membrane depolarization during diastole and then the heart rate,
are still a matter of debate. (DiFrancesco, 1993)
argues that the hyperpolarization activated current (if ) is the only current that can generate and
control the slow depolarization of pacemaker cells.
This current is normally carried by N a+ and K + .
(Guo et al., 1995) reported another current, called
the sustained inward current ist , where the major
charge carrier is believed to be N a+ . Also a Ca2+
“window” current has been observed in rabbit
sinoatrial node cells (Denyer and Brown, 1995).
It is possible that any of these currents, or a
combination of them, is responsible for membrane
depolarization during diastole.
During diastole the electrochemical driving forces
produce outward K + currents and inward N a+
and Ca2+ currents, and the driving force for Ca2+
is much larger than that for N a+ (Endresen et al.,
2000). This implies that a significant background
influx of Ca2+ is possible during diastole, and that
this current might be responsible for pacemaking
activity in sinoatrial node cells (Boyett et al.,
2001). The conductance for this current is denoted
I¯b,Ca and used in the proposed model to control
the voltage-dependent calcium current ICa,t . I¯b,Ca
is responsible for the slow diastolic depolarization,
and appears to have control capability of the
pacemaker activity. In fact, changes in the cycle
length of the pacemaker AP will be observed in
an almost linear relationship with I¯b,Ca . Then,
we conclude that the autonomous nervous system
interacts almost linearly in the regulation of heart
rate as assumed in (Warner and Russell, 1969).
JCT
Sarcoplasmic
Reticulum
SR
2+
Ca
2+
Mg
+
Na
Troponin Ca
Troponin Mg
+
Na−K
pump
Inak
Calcium Buffers
Ca
Troponin C
Inaca
+
Na−Ca
exchanger
2+
Ena Ek
Eca
Ina,t Ik,t Ica,t
Ik,t
Junctional
Ina,t
Na
External
Space
Ica,t
Calsequestrin
Bulk Cytosol
Cm
K
In this paper, a model for a cardiac pacemaker
cell is proposed. It is realistic enough to exhibit many of the characteristics of larger pacemaker cell models, and yet, is simple enough
with only ten state variables. It has furthermore
a sound asymptotic behaviour without drifts of
the state, a useful property for multi-beat simulations. The same model structure has been
used in (Djabella and Sorine, 2005) to represent
excitation–contraction coupling in a ventricular
cell. It consists of two parts: 1) a cell membrane
with capacitance, voltage-dependent ion channels,
electrogenic pump and exchanger, the ionic currents model being derived using conservation laws
as in (Endresen et al., 2000), and 2) a lumped
compartmental model that accounts for intracellular changes in concentrations of N a+ , K +
(Hund et al., 2001) and the main processes that
regulate intracellular calcium concentration: release and uptake by the sarcoplasmic reticulum
(SR), buffering in the SR (Tusscher et al., 2004)
and in the bulk cytosol (Shannon et al., 2004).
Intracellular
Space
Calmodulin
Sarcolemma
Fig. 1. A representation of the model of a cardiac
pacemaker cell: electrical equivalent circuit
for the sarcolemma and fluid compartment
The paper is organized as follows: the model is
described in Section 2. Section 3 shows some
simulation results. A discussion and conclusions
are presented in Sections 4 and 5 respectively.
2. MODEL DESCRIPTION
The mathematical model is a nonlinear system of
ten first order ordinary differential equations. The
detailed equations, parameters values and abbreviations used are presented in the APPENDIX.
Figure 1 shows the lumped electrical equivalent
circuit for the sarcolemma and the fluid compartment system of a single pacemaker cardiac
cell. The membrane model includes both the
potential–mediated ion channels responsible for
the dynamic aspects of the membrane AP (inward
sodium, L-type calcium and outward potassium)
and N a+ /Ca2+ exchangers, N a+ /K + pumps.
The fluid compartment is modelled by seven
differential equations, three of them describing
the intracellular concentrations of free Calcium,
Sodium and Potassium. The remaining four equations describe the binding of Ca2+ to specific
sites on the myoplasmic troponin and calmodulin proteins, taking into account the competition
between Ca2+ and M g 2+ . The Ca2+ buffering
system is very important for the regulation and
limitation of free intracellular Ca2+ concentration
transients.
3. RESULTS
The dynamic behavior of the cell model was computed by solving the system of nonlinear ordinary
differential equations with a second order modified
Rosenbrock method with variable steplength. The
initial conditions are listed in the table 1.
Concentrations (mM)
0.02
(pA)
0
Ca,t
I
1
Cai
Ca
−200
SR
0.5
−400
−600
−800
0
0.5
1
1.5
0
2
Occupancy
IK,t (pA)
400
200
0
0
0.5
1
1.5
2
0.5
1
1.5
1
2
θ
θ
Tn
TnCa
0.5
θ
Cal
0
0.5
1
1.5
2
0.5
1
1.5
2
1
1.5
2
Na (mM)
0
−4
−6
0
0.5
1
1.5
21.6
21.55
0
2
127.4
12
11
10
9
8
I
NaK
400
127.3
i
Currents (pA)
21.65
i
−2
K (mM)
INa,t (pA)
21.7
I
NaCa
0
0.5
1
1.5
2
−600
127.2
0
0.5
Time (s)
Time (s)
Fig. 3. Computed spontaneous AP and ionic currents (left) and intracellular Ca2+ dynamics (right)
2
0.01
1.8
Pacemaker frequency (Hz)
0
−0.01
V (V)
−0.02
−0.03
−0.04
−0.05
−0.06
−0.07
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Fig. 2. Spontaneous action potential
1.2
1
5
5.5
6
6.5
7
Ib,Ca (pA)
7.5
8
8.5
9
−3
x 10
Fig. 5. The relationship between the pacemaker
frequency and the amplitude of the pacemaker current: the control effect
0.02
Reduced I
b,Ca
0.01
Control
zero potentials
0
−0.01
−0.02
V (V)
1.4
0.8
Time (s)
−0.03
−0.04
−0.05
−0.06
−0.07
−0.08
0
1.6
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Time (s)
Fig. 4. Effect of a reduced I¯b,Ca on action potential
Figure 2 shows the computed AP that had a cycle
length, amplitude, duration (measured at -30 mV)
and maximal diastolic potential of 550ms, 85mV ,
120ms and −71mV , respectively. This simulated
pacemaker AP has a reasonable shape compared
with those recorded experimentally.
Figure 3 shows, on the left, the computed temporal behaviour of sarcolemmal ionic currents. On
the right, it shows the intracellular Ca2+ dynamics including the changes in Ca2+ concentration in
the SR, the associated changes in the occupancy
ratio of Ca2+ buffers and the changes in N ai
and Ki during pacemaker activity. These computed waveform changes during spontaneous AP
are very similar to those recorded experimentally
(Demir et al., 1994).
In Fig. 4, the AP computed under control conditions is compared with that computed when the
pacemaker current I¯b,Ca was affected. It shows:
- a cycle length increase from 550 to 900ms,
- an AP amplitude slight change from 85 to 88mV ,
- an AP duration increase, at −30mV , from 120
to 130ms, and
- a maximal diastolic potential decrease of 5mV .
These changes are qualitatively similar to those
observed experimentally in rabbit sinoatrial node
cells (Boyett et al., 2001).
Reduced I
b,Ca
Control
(pA)
Ca,t
−200
0.005
I
Cai (mM)
0
0.01
0
0
1
−400
−600
−800
0
2
1
2
400
IK,t (pA)
−5
I
Na,t
(pA)
0
−10
0
1
100
1
2
1
2
1000
(pA)
11
NaCa
10.5
10
9.5
500
0
I
(pA)
NaK
I
200
0
0
2
11.5
9
0
300
1
2
Time (s)
−500
0
Time (s)
Fig. 6. Ca2+ transients and ionic currents computed under control conditions and for a reduced I¯b,Ca
Figure 5 shows the relationship between the pacemaker frequency and the amplitude of the pacemaker current I¯b,Ca : it appears to be approximately linear. This has an important consequence:
there is no need for a more complex model for
quantitative simulations of pacemaker under the
control of the autonomic nervous system. This
frequency control can be described fairly well by
assuming that the sympathetic–parasympathetic
balance interacts almost linearly on the heart rate
as assumed in (Warner and Russell, 1969).
Figure 6 shows the effect of a reduced pacemaker
current I¯b,Ca on the simulated intracellular calcium concentration and ionic currents. In this
case, there is no slight change in the Ca2+ transients shapes or in transmembrane ionic currents
waveforms (there is a small decrease in the IN a,t
current amplitude): they were only shifted. In
conclusion, only pacemaking rate was affected. In
the present paper ICa,t means ICa,L (Djabella and
Sorine, 2005) and the I¯b,Ca affects this current (eq.
6), therefore, the simulations are consistent with
the experimental data (Boyett et al., 2001).
4. DISCUSSION
A simple model for a cardiac pacemaker cell
has been presented. It involves only N a+ , K +
and Ca2+ ions, their respective channels, the
N a+ /Ca2+ exchanger, and the N a+ /K + pump.
It also includes a description of the dynamics of
the main calcium buffers in the bulk cytosol and
in the SR, and it takes into account the calcium
uptake and release from SR.
The model is able to produce sinoatrial node AP,
the behavior of the most important currents and
the intracellular Ca2+ dynamics (concentration
changes, SR Ca2+ uptake and release, and Ca2+
buffering) involved during normal pacemaking.
The contribution of the currents to the pacemaker
activity is still ill known despite many studies
(Rasmusson et al., 1990; Demir et al., 1994; Kurata et al., 2002). All of them agree that the calcium plays an important role in the slow depolarization of the cardiac pacemaker potential. In recent years much attention has been focused on the
possibility that Ca2+ can control the pacemaker
activity of the sinoatrial node. In this article, using
computer simulations, it has been shown that a
significant background influx of Ca2+ may be important in the changes in the AP and pacemaker
activity (Boyett et al., 2001).
The proposed model and simulations can provide
guidance for future developments of more realistic
controlled cardiac pacemaker cell for use in models
of the intact sinoatrial node or in whole heart
models in three dimensions.
5. CONCLUSION
A differential model of cardiac pacemaker cell has
been presented. Simulations of the electrical pacemaker activity and associated cytosolic ion concentration changes appear realistic. They demonstrate the control of the pacemaker activity by the
calcium, so that the modulation of this activity by
the autonomous nervous system can be explained
fairly well by assuming that the sympathetic–
parasympathetic system interacts linearly on the
heart rate via a calcium input.
Due to its sound asymptotic behavior without
drifts of the state and to its medium complexity,
this model can provide guidance for future modelling work on the controlled cardiac pacemaker
cells in multi-beat simulations from the cell to the
heart scales.
6. APPENDIX
The proposed model of cardiac pacemaker cell is
the following system of differential equations:
IK,t +IN a,t +ICa,t +IN aK +IN aCa
dV
=−
dt
Cm
dKi 2IN aK − IK,t
=
dt
F VC
IN a,t + 3IN aK + 3IN aCa
dN ai
=−
dt
F VC
dCai 2IN aCa −ICa,t
=
+Jleak +Jrel −Jup
dt
2F V
X dθb C
, IB = {T n, Cal, T nCa}
Bb
−
dt
b∈IB
dgX gX∞ − gX
(1)
=
, X ∈ {N a, K}
dt
τg X
dθT n
= kTonn |Cai |+ (1 − θT n ) − kTofnf θT n
dt
dθCal
of f
on
= kCal
|Cai |+ (1 − θCal ) − kCal
θCal
dt
dθT nCa
= kTonnCa |Cai |+ (1 − θT nCa − θT nM g )
dt
f
−kTofnCa
θT nCa
dθT nM g
on
= kT nM g |M gi |+ (1 − θT nCa − θT nM g )
dt
f
−kTofnM
g θT nM g
The gate dynamics are defined by
1
V − V gX
gX∞ = 1+tanh
,
2
RT /2F
τX
, X ∈ {N a, K, d, m}
τg X =
V −VgX
cosh RT /2F
(2)
where X = d, m represent fast Ca, N a activation
gating, denoted as usual d∞ = gd∞ , m∞ = gm∞ .
Setting
Xe RT
log , X ∈ {Ca, N a, K} , (3)
VX =
zX F
Xi
the currents through the membrane are then:
V −VK
IK,t = I¯K gK sinh 2RT
/F
IN a,t = I¯N a gN a m∞ sinh V −VN a
(4)
(5)
2RT /F
ICa,t = [I¯Ca (1−gK )d∞+I¯b,Ca ] sinh
N a −VAT P
IN aK = I¯N aK tanh V +2VK −3V
2RT /F
Ca −3VN a
IN aCa = I¯N aCa sinh V +2V2RT
/F
V −VCa
RT /F
(6)
(7)
(8)
Finally, the CICR mechanism is described using
Cai H CaSR H
Kmf − Kmr Jup = Qup Jmax
i H CaSR H
+
1 + KCa
Kmr mf
(9)
Jrel = Krel d∞ (CaSR − CaiJCT )
(10)
Jleak = Kleak (CaSR − CaiJCT )
(11)
BJCT |Cai |+
CaiJCT =
(12)
|Cai |+ + KJCT
BSR |CaSR |+
=
CaSR +
|CaSR |+ + KSR
!
X
VC
Bb θb −CaiJCT (13)
CaT −Cai −
VSR
b∈IB
CaT =
Vext
Cm
(V − Vext ) − N ai − Ki
F VC
F VC
=−
(N ae + Ke + 2Cae )
Cm
1
2
(14)
(15)
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Boyett, M. R., H. Zhang, A. Garny and A. V.
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Table 1 Initial conditions
Variables
V0
Ki0
N ai0
Cai0
θT n0
θT nCa0
θT nM g0
θCal0
gK0
gN a0
Initial values
−51.4 10−3
127.8
20.7
0.001
0.0451
0.139
0.0003
0.0031
0.074
0.029
Units
V
mM
mM
mM
Table 2 Abbreviations used in the text
Abbrev.
V
Vext
IX,t
IN aCa
IN aK
gK
gN a
d∞
m∞
Jrel
Jup
Jleak
Xi
Xe
CaT
CaSR
θT n
θT nCa
θT nM g
θCal
CaiJCT
Definitions
Action potential
External stimulus voltage
Total X current through all channels
N a+ − Ca2+ exchanger current
N a+ − K + pump current
Potassium activation gating
Sodium activation gating
Fast Calcium activation gating
Fast Sodium activation gating
Calcium-Induced Calcium Release current
Pump current taking up calcium in the SR
Leakage current from SR to the cytoplasm
Intracellular concentration of the free ion X
External concentration of the ion free X
Total calcium concentration in the cell
Free calcium concentration in the SR
Fraction of Troponin–C sites bound with Ca
Fraction of Troponin–Ca sites bound with Ca
Fraction of Troponin–Mg sites bound with Mg
Fraction of Calmodulin sites bound with Ca
SR Calcium buffered in the junction
Table 3 Model parameters
Parameter
VC
VSR
F
Vm
Vd
VgK
VgN a
τN a = τK
VAT P
Cm
R
T
zN a = zK
zCa
I¯N a
I¯K
I¯Ca
I¯b,Ca
I¯N aK
I¯N aCa
Ke
N ae
Cae
M gi
Kleak
H
Kmf
Kmr
Jmax
Qup
Krel
on
kT
n
of f
kT
n
BT n
on
kT
nCa
of f
kT
nCa
BT nCa
on
kT
nM g
of f
kT
nM g
BT nM g
on
kCal
of f
kCal
BCal
KJCT
BJCT
KSR
BSR
Value
16.404
1.094
96.486
−56.86 10−3
−5 10−3
−26 10−3
−71.55 10−3
0.2
−450 10−3
22
8.314 10−3
310
1
2
11.27
164.5
131
0.0074
11.46
7000
5.4
140
2
1
5 10−8
1.787
0.246 10−3
1.7
286 10−3
2.6 10−5
25 10−2
32.7 103
19.6
70 10−8
2.37 103
0.032
140 10−8
0.003 103
Unit
pL
pL
Cmmol−1
V
V
V
V
s
V
pF
Jmmol−1 K −1
K
3.33
140 10−3
34 103
238
24 10−8
13 10−3
4.6 10−3
0.0065
0.14
s−1
mM
mM −1 s−1
s−1
mM
mM
mM
mM
mM
pA
pA
pA
pA
pA
pA
mM
mM
mM
mM
s−1
mM
mM
mM s−1
s−1
mM −1 s−1
s−1
mM
mM −1 s−1
s−1
mM
mM −1 s−1
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